Testing Gaussian Process with Applications to Super-Resolution

This article introduces new testing procedures on the mean of a stationary Gaussian process. Our test statistics are exact and derived from the outcomes of total variation minimization on the space of complex valued measures. Two testing procedures are presented, the first one is based on thin grids (we show that this testing procedure is unbiased) and the second one is based on maxima of the Gaussian process. We show that both procedures can be performed even if the variance is unknown. These procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented.